This paper is concerned with the existence of constant scalar curvature Kähler metrics on blow-ups at finitely many points of compact manifolds which already carry constant scalar curvature Kähler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature Kähler metrics.

We introduce a curvature-dimension condition CD ($K$,$N$) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$ (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD($K$, ∞). The additional real parameter$N$plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD($K$,$N$) is equivalent to $${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $$ and dim($M$) ⩽$N$.

We introduce and analyze lower ($Ricci$) curvature bounds $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$for metric measure spaces $ {\left( {M,d,m} \right)} $ . Our definition is based on convexity properties of the relative entropy $ Ent{\left( { \cdot \left| m \right.} \right)} $ regarded as a function on the$L$_{2}-Wasserstein space of probability measures on the metric space $ {\left( {M,d} \right)} $ . Among others, we show that $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$implies estimates for the volume growth of concentric balls. For Riemannian manifolds, $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$if and only if $ Ric_{M} {\left( {\xi ,\xi } \right)} $ ⩾$K$ $ {\left| \xi \right|}^{2} $ for all $ \xi \in TM $ .

We give an explicit construction of a family of lattices in PU (1, 2) originally constructed by Livné. Following Thurston, we construct these lattices as the modular group of certain Euclidean cone metrics on the sphere. We give connections between these groups and other groups of complex hyperbolic isometries.

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